Question: Lucia and Maria are business women who decided to invest money by buying farm land in Brazil. Lucia bought $11$ hectares of land in the first month, and each month afterwards she buys $5$ additional hectares. Maria bought $6$ hectares of land in the first month, and each month afterward her total number of hectares increases by a factor of $1.4$. They started their investments at the same time, and they both buy the additional land at the beginning of each month. What is the first month in which Maria's amount of land exceeds Lucia's amount of land?
Answer: Notice that Lucia's total grows linearly while Maria's total grows exponentially. This means Maria's total is bound to exceed Lucia's total at some point. Let's start calculating each woman's total amount of land to see when that happens. Month Lucia Maria (Add $5$ each month.) (Multiply by $1.4$ each month.) $1$ $11$ $6$ $2$ $16$ $8.4$ $3$ $21$ $11.76$ $4$ $26$ $16.46$ $5$ $31$ $23.05$ $6$ $36$ $32.27$ $7$ $41$ $45.18$ In conclusion, Maria's amount of land will first exceed Lucia's amount of land in month number $7$.